Subgroups — #17034709

Order 4352.

1350 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762833	C₂	1	6	yes	maximal
2	#14528936	C₂	1	8	yes	maximal, complemented
3	C₂×C₄³×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63223	C₂²	1	4	yes	—
5	#11762833	—	2	24	no	—
6	C₂×C₄⋊#3622	C₂²	1	24	yes	—
7	C₂²×C₄²×C₁₇	C₄	1	4	yes	—
8	C₂²×C₄²×C₁₇	C₂²	1	3	yes	—
9	C₄³×C₁₇	C₄	1	3	yes	—
10	C₄³×C₁₇	C₄	1	1	yes	characteristic
11	C₄³×C₁₇	C₂²	1	4	yes	—
12	C₂³×C₄×C₁₇	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₇	—	2	16	no	—
15	C₂×C₄²×C₁₇	C₂×C₄	1	12	yes	—
16	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
17	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
18	C₂×C₄²×C₁₇	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₁₇	D₄	1	12	yes	—
20	C₂×C₄²×C₁₇	Q₈	1	12	yes	—
21	C₂×C₄²×C₁₇	C₂³	1	6	yes	—
22	C₂×#63223	—	2	24	no	—
23	C₂²×#3622	C₂³	1	8	yes	—
24	C₄⋊#3622	—	2	48	no	—
25	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	—	2	24	no	—
27	C₂²×C₄×C₁₇	C₄⋊C₄	1	24	yes	—
28	C₂²×C₄×C₁₇	C₂²×C₄	1	6	yes	—
29	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂²×C₄	1	6	yes	—
31	C₂²×C₄×C₁₇	C₂×D₄	1	6	yes	—
32	C₂²×C₄×C₁₇	C₂×Q₈	1	6	yes	—
33	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
34	C₄²×C₁₇	—	2	24	no	—
35	C₄²×C₁₇	C₄⋊C₄	1	12	yes	—
36	C₄²×C₁₇	C₄⋊C₄	1	12	yes	—
37	C₄²×C₁₇	M₁₆	1	16	yes	complemented
38	C₄²×C₁₇	C₂×D₄	1	12	yes	—
39	C₄²×C₁₇	C₂×Q₈	1	12	yes	—
40	#63223	—	4	16	no	—
41	C₂×#3622	—	2	24	no	—
42	#20791	—	17	1	no	maximal, Sylow
43	C₂³×C₁₇	—	2	4	no	—
44	C₂³×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
45	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	—	2	12	no	—
47	C₂×C₄×C₁₇	—	2	48	no	—
48	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	12	yes	—
49	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
50	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	12	yes	—
51	C₂×C₄×C₁₇	#127	1	4	yes	—
52	C₂×C₄×C₁₇	C₄⋊Q₈	1	12	yes	—
53	C₂×C₄×C₁₇	C₂×M₁₆	1	24	yes	—
54	C₂×C₄×C₁₇	C₂²×D₄	1	3	yes	—
55	C₂×C₄×C₁₇	C₂²×Q₈	1	3	yes	—
56	#3622	—	4	8	no	—
57	C₂×#423	—	17	6	no	—
58	C₂×C₄³	D₁₇	1	1	yes	characteristic
59	#1842	—	17	8	no	—
60	C₂²×C₁₇	—	2	12	no	—
61	C₂²×C₁₇	#384	1	4	yes	—
62	C₂²×C₁₇	C₂²×C₄⋊C₄	1	3	yes	—
63	C₂²×C₁₇	C₂²×M₁₆	1	4	yes	—
64	C₄×C₁₇	—	2	12	no	—
65	C₄×C₁₇	#384	1	4	yes	—
66	C₄×C₁₇	#423	1	24	yes	complemented
67	C₄×C₁₇	C₂×#127	1	1	yes	characteristic
68	C₄×C₁₇	C₂×C₄⋊Q₈	1	3	yes	—
69	C₂²×M₁₆	—	17	4	no	—
70	C₂²×C₄²	Dic₁₇	1	4	yes	—
71	C₂²×C₄²	C₂×D₁₇	1	3	yes	—
72	#423	—	34	24	no	—
73	C₂×C₄⋊C₈	—	17	24	no	—
74	C₄³	Dic₁₇	1	3	yes	—
75	C₄³	Dic₁₇	1	1	yes	characteristic
76	C₄³	C₂×D₁₇	1	4	yes	—
77	C₂×C₁₇	—	2	4	no	—
78	C₂×C₁₇	C₂×#384	1	1	yes	characteristic
79	C₂×C₁₇	C₂×#423	1	6	yes	—
80	C₂³×C₄	C₂×Dic₁₇	1	6	yes	—
81	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
82	C₂×M₁₆	—	34	24	no	—
83	C₂²×C₈	—	17	8	no	—
84	C₂×C₄²	—	2	16	no	—
85	C₂×C₄²	Dic₃₄	1	12	yes	—
86	C₂×C₄²	D₆₈	1	12	yes	—
87	C₂×C₄²	C₂×Dic₁₇	1	12	yes	—
88	C₂×C₄²	C₂×Dic₁₇	1	4	yes	—
89	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
90	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
91	C₂×C₄²	C₂²×D₁₇	1	6	yes	—
92	C₄⋊C₈	—	34	48	no	—
93	C₁₇	#20791	1	1	yes	characteristic, complemented, Sylow
94	C₂⁴	C₂²×Dic₁₇	1	1	yes	characteristic
95	C₂²×C₄	—	2	24	no	—
96	C₂²×C₄	C₄⋊Dic₁₇	1	24	yes	—
97	C₂²×C₄	C₂×Dic₃₄	1	6	yes	—
98	C₂²×C₄	C₂×D₆₈	1	6	yes	—
99	C₂²×C₄	C₂²×Dic₁₇	1	6	yes	—
100	C₂²×C₄	C₂²×Dic₁₇	1	6	yes	—
101	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
102	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
103	M₁₆	—	68	16	no	—
104	C₂×C₈	—	34	24	no	—
105	C₄²	—	2	24	no	—
106	C₄²	#63223	1	16	yes	complemented
107	C₄²	C₄⋊Dic₁₇	1	12	yes	—
108	C₄²	C₄⋊Dic₁₇	1	12	yes	—
109	C₄²	C₂×Dic₃₄	1	12	yes	—
110	C₄²	C₂×D₆₈	1	12	yes	—
111	C₂³	—	2	4	no	—
112	C₂³	C₂×C₄⋊Dic₁₇	1	6	yes	—
113	C₂³	C₂³×Dic₁₇	1	1	yes	characteristic
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	—	2	48	no	—
116	C₂×C₄	#10587555	1	12	yes	—
117	C₂×C₄	#10587560	1	4	yes	—
118	C₂×C₄	C₂×#63223	1	24	yes	—
119	C₂×C₄	C₂×C₄⋊Dic₁₇	1	12	yes	—
120	C₂×C₄	C₂×C₄⋊Dic₁₇	1	12	yes	—
121	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
122	C₂×C₄	C₂²×Dic₃₄	1	3	yes	—
123	C₂×C₄	C₂²×D₆₈	1	3	yes	—
124	C₈	—	68	8	no	—
125	C₂²	—	2	12	no	—
126	C₂²	#11762679	1	4	yes	—
127	C₂²	C₂²×C₄⋊Dic₁₇	1	3	yes	—
128	C₂²	C₂²×#63223	1	4	yes	—
129	C₄	—	2	12	no	—
130	C₄	#11762679	1	4	yes	—
131	C₄	#11762833	1	24	yes	complemented
132	C₄	C₂×#10587560	1	1	yes	characteristic
133	C₄	C₂×#10587555	1	3	yes	—
134	C₂	—	2	4	no	—
135	C₂	C₂×#11762679	1	1	yes	characteristic
136	C₂	C₂×#11762833	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₇
commutator, socle	C₂³×C₁₇	C₂³×C₄
Fitting	C₂×C₄³×C₁₇	C₂
prime radical	C₂⁴×C₁₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₄³	D₁₇
hypocenter	C₁₇	#20791
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #14528936	C₂	semidirect
37. C₄²×C₁₇	M₁₆	semidirect
66. C₄×C₁₇	#423	semidirect
93. C₁₇	#20791	semidirect
106. C₄²	#63223	semidirect
131. C₄	#11762833	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11762833	2
#14528936	2
C₂×C₄³×C₁₇	2
#20791	17